Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the area of the region between the two concentric circles \(r=7\) and \(r=10\)

Step-by-Step Solution

Answer

The area is \(51\pi\).

Step 1: Understand the Problem

We are given two concentric circles with radii 7 and 10. We need to find the area of the region between these two circles. This area forms an annulus, which resembles a 'ring' between the two circles.

Step 2: Formula for Area of a Circle

Recall the formula for the area of a circle: \[ A = \pi r^2 \]where \( r \) is the radius of the circle.

Step 3: Calculate Area of the Larger Circle

Using the formula from Step 2, calculate the area of the larger circle with radius 10.\[ A_{\text{large}} = \pi (10)^2 = 100\pi \]

Step 4: Calculate Area of the Smaller Circle

Next, calculate the area of the smaller circle with radius 7 using the same formula.\[ A_{\text{small}} = \pi (7)^2 = 49\pi \]

Step 5: Find the Area of the Annulus

Subtract the area of the smaller circle from the area of the larger circle to find the area of the annulus:\[ A_{\text{annulus}} = A_{\text{large}} - A_{\text{small}} = 100\pi - 49\pi = 51\pi \]

Step 6: Final Answer

The area of the region between the two concentric circles is \( 51\pi \).

Key Concepts

Annulus

An annulus is a geometric figure that looks like a ring. It is the space between two concentric circles, which means these circles share the same center point. You can imagine an annulus as the shape you get when you cut out a smaller circle from a larger one. In this exercise, the annulus is formed between two circles with different radii: 7 and 10. The area of an annulus is what remains after removing the inner circle's area from the larger circle's area. This concept of an annulus is essential in various fields such as architecture, engineering, and design. It's crucial because it often represents areas like washers and circular bands. To find the area of an annulus, we need to calculate the area of the larger circle and subtract the area of the smaller circle.

Circle Area Formula

The area of a circle is calculated with a simple and powerful formula:

• \[ A = \pi r^2 \]

where \( A \) is the area and \( r \) is the radius of the circle. The symbol \( \pi \) (pi) is a mathematical constant approximately equal to 3.14159. It represents the ratio of the circumference of any circle to its diameter. To find the area of a circle, you multiply \( \pi \) by the square of the radius. This formula allows us to easily compute circles' areas in various applications, from calculating the space inside a tire to determining the size of a pizza. In this particular task, you will use this formula twice: once for the larger circle with a 10-unit radius and once for the smaller circle with a 7-unit radius. Using this foundational formula correctly is important to understanding and visualizing the size of circular shapes.

Subtraction of Areas

Subtraction of areas is a straightforward process usual in geometry, particularly when dealing with composite shapes. It involves identifying and subtracting one area from another to find the remaining space. In the case of the annulus, the goal is to subtract the area of the inner, smaller circle from the area of the larger circle. This gives you the area of the ring-shaped region, or the annulus.For our problem, we calculated:

• The area of the larger circle: \[ 100\pi \]
• The area of the smaller circle: \[ 49\pi \]

By subtracting these two areas,

• \( A_{\text{annulus}} = 100\pi - 49\pi = 51\pi \)

we find that the area of the annulus is \( 51\pi \). This process of subtraction lets us see how much 'material' the ring contains. This principle is widely used for calculating areas that are not straightforward, helping us solve real-world problems by breaking them into simpler parts. So, whenever you encounter a ring-shaped object, think of subtraction of areas to find the solution!